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Find the derivative of $g(x)$ . $\newline$ $g(x) = \ln(2x)$ $\newline$ $g^{\prime}(x) =$ ______

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Find derivatives using the quotient rule I

Full solution

Q. Find the derivative of

g(x)

.

\newline

g(x) = \ln(2x)

\newline

g^{\prime}(x) =

______

Identify function to differentiate: Identify the function to differentiate. The function is $g(x) = \ln(2x)$ .
Apply chain rule for differentiation: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. The outer function is $\ln(u)$ with $u = 2x$ , and the derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$ . The inner function is $2x$ , and its derivative with respect to $x$ is $2$ .
Calculate derivative of inner function: Calculate the derivative of the inner function. The derivative of $2x$ with respect to $x$ is $2$ .
Apply chain rule using derivatives: Apply the chain rule using the derivatives from steps $2$ and $3$ . The derivative of $g(x)$ is $\left(\frac{1}{2x}\right) \cdot 2$ .
Simplify the expression: Simplify the expression. Multiplying $\frac{1}{2x}$ by $2$ gives us $\frac{1}{x}$ .
Write final answer: Write the final answer. The derivative of $g(x) = \ln(2x)$ is $g'(x) = \frac{1}{x}$ .

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